--- title: Gromoll–Meyer Sphere url: https://www.emergentmind.com/topics/gromoll-meyer-sphere type: topic --- # Gromoll–Meyer Sphere Searching arXiv for recent and foundational papers on the Gromoll–Meyer sphere, especially the provided 2024 paper and closely related geometric works. The Gromoll–Meyer sphere, usually denoted \(\Sigma^7_{GM}\), is an exotic \(7\)-sphere: a smooth manifold homeomorphic but not diffeomorphic to the standard sphere \(S^7\). It is realized as a quotient of \(\mathrm{Sp}(2)\) by a free \(\mathrm{Sp}(1)\cong S^3\)-action, and it occupies a singular position in differential geometry as the first exotic sphere known to admit nonnegative sectional curvature and, in the biquotient description, as the only exotic sphere expressible as a biquotient of a compact Lie group [1908.01990]. It has subsequently served as a model object for positive-curvature constructions, transnormal and sub-Riemannian structures, equivariant quotient constructions, and Lorentzian and Kaluza–Klein geometries [0805.0812]. ## 1. Classical quotient construction and exoticity A standard quaternionic description starts from \[ \mathrm{Sp}(2)=\left\{\begin{pmatrix} a & c\\ b & d\end{pmatrix}\in \mathrm{S}^7\times \mathrm{S}^7~ \Big| ~a\overline{b} + c\overline{d} = 0\right\}, \] with \(a,b,c,d\in\mathbb H\). In this model, the first-row projection \[ \pi:\mathrm{Sp}(2)\to \mathrm{S}^7 \] is a principal \(\mathrm{SU}(2)\)-bundle, where \(\mathrm{SU}(2)\cong \mathrm{Sp}(1)\) is the group of unit quaternions. The principal action \(\bullet\) is \[ \begin{pmatrix} a & c \\ b & d \end{pmatrix}\overline q = \begin{pmatrix} a & c\overline{q}\\ b & d\overline{q} \end{pmatrix}, \] while the Gromoll–Meyer action \(\star\) is \[ q \begin{pmatrix} a & c \\ b & d \end{pmatrix} = \begin{pmatrix} qa\overline{q} & qc \\ qb\overline{q} & qd \end{pmatrix}. \] Its quotient is the Gromoll–Meyer exotic sphere, with quotient map \[ \pi'\left( \begin{pmatrix} a & c \\ b & d \end{pmatrix}\right) := \left(2\overline cd,|c|^2-|d|^2\right), \] so that \[ \Sigma^7_{GM}=\mathrm{Sp}(2)/\star,\qquad \mathrm{S}^7=\mathrm{Sp}(2)/\bullet. \] The resulting pair \(\mathrm{S}^7\) and \(\Sigma^7_{GM}\) are repeatedly emphasized as “homeomorphic but not diffeomorphic manifolds” arising from the same total space by two commuting \(\mathrm{SU}(2)\)-actions [2403.08960]. A closely related formulation treats \(\Sigma^7_{GM}\) as a biquotient. In that language, the quotient is written as \(\mathrm{Sp}(2,\mathbb H)//\mathrm{Sp}(1,\mathbb H)\), with the \(\mathrm{Sp}(1)\)-action acting on the left by \(\mathrm{diag}(q,q)\) and on the right by \(\mathrm{diag}(q,1)\) [1908.01990]. This biquotient presentation is not merely notational: it underlies the metric, curvature, and equivariant constructions that dominate the later literature. ## 2. Commuting actions, cross-diagrams, and pullback models The quotient construction is most naturally organized by a commuting-action diagram. In the language of \(\star\)-diagrams, if \(G\hookrightarrow P\to M\) is a principal bundle with principal action \(\bullet\), and there is another free action \(\star\) of the same compact connected Lie group \(G\) on \(P\) commuting with \(\bullet\), then one obtains \[ M=P/\bullet,\qquad M'=P/\star. \] The Gromoll–Meyer example is the motivating model with \[ G=\mathrm{SU}(2),\qquad P=\mathrm{Sp}(2),\qquad M=\mathrm{S}^7,\qquad M'=\Sigma^7_{GM}, \] and it is treated as the paradigmatic instance of a construction in which local orbit geometry is equivariantly related while the global smooth structures differ [2403.08960]. This viewpoint was abstracted further in the pullback and cross-diagram framework. The original Gromoll–Meyer construction is encoded by \[ S^3\stackrel{\bullet}{\cdots} Sp(2)\stackrel{pr}{\to} S^7, \] together with the second free action \[ q\star\begin{pmatrix}a & c\\ b & d\end{pmatrix} = \begin{pmatrix}qa\bar q & qc\\ qb\bar q & qd\end{pmatrix}, \] whose quotient is the exotic sphere \(\Sigma^7\). In local trivializations, \(Sp(2)\) with both actions is equivariantly diffeomorphic to \[ D^4\times S^3\times S^3\cup_{f_\theta}S^3\times D^4\times S^3, \qquad f_\theta(x,y,g)=(x,y,gx\bar y), \] so that the clutching function of \(Sp(2)\to S^7\) is \[ \theta(x,y)=x\bar y. \] This cross-diagram mechanism was pulled back along equivariant maps to construct analogous quotient models for exotic \(8\)- and \(10\)-spheres, with total spaces \(E^{11}=f_8^*Sp(2)\) and \(E^{13}=f_{10}^*Sp(2)\), and quotients diffeomorphic to the only exotic sphere of dimension \(8\) and a generator of the index \(2\) subgroup of \(10\)-dimensional homotopy spheres [1010.6039]. The same pullback perspective also clarifies what is structural in the original \(7\)-dimensional example. The Gromoll–Meyer sphere is not treated as an isolated biquotient accident, but as the prototype of a wider bundle-plus-commuting-action mechanism in which exoticity is encoded equivariantly [1010.6039]. ## 3. Riemannian metrics, curvature, and extrinsic realizations For the bi-invariant metric on \(\mathrm{Sp}(2)\), the quotient projection to \(\Sigma^7\) is a Riemannian submersion, so \(\Sigma^7\) carries a metric of nonnegative sectional curvature. This is the classical Gromoll–Meyer result, and the sphere is stressed as the first exotic sphere with nonnegative sectional curvature [1912.02431]. Later, Petersen and Wilhelm proved that there is a metric on the Gromoll–Meyer sphere with positive sectional curvature [0805.0812]. A complementary intrinsic analysis studies a \(2\)-parameter family of left-invariant metrics \(g_r=g_{(r_1,r_2)}\) on \(\mathrm{Sp}(2)\), defined by \[ \left|Q\begin{pmatrix} x & y\\ -\bar y & z \end{pmatrix}\right|^2 = \frac{r_1}{2}|x|^2+|y|^2+\frac{r_2}{2}|z|^2. \] For this family, \[ g_r \text{ has nonnegative sectional curvature iff } r_1+r_2\le 2, \] and \[ g_r \text{ is Einstein iff } r_1=r_2=1 \quad\text{or}\quad r_1=r_2=2. \] In an extrinsic extension of the quotient picture, copies of \(\mathrm{Sp}(2)\) appear as isoparametric hypersurfaces \(Sp(2)_\theta\subset \widetilde N^{11}\), and their quotients \(\Sigma^7_\theta\subset N^8\) are all diffeomorphic to the Gromoll–Meyer sphere. For each \(\theta\in(0,\pi)\), the induced metric on \(\Sigma^7_\theta\) has positive Ricci curvature and quasi-positive sectional curvature simultaneously [1912.02431]. A separate Kaluza–Klein Ansatz realizes the Gromoll–Meyer sphere as the Milnor \((m,n)=(2,1)\) \(S^3\)-bundle over \(S^4\), with a round \(S^4\) as base, unit \(S^3\) as fibre, and \(k=2\) and \(k=1\) \(SU(2)\) instantons as gauge fields. At a distinguished point \[ a=\frac1{\sqrt3},\qquad \lambda=\frac2{\sqrt3} \] in the equal-size \(k=2\) instanton moduli, combined with the maximally symmetric \(k=1\) instanton, the resulting metric has maximal isometry \[ SO(3)\times O(2). \] For this family the Ricci tensor is computed explicitly, and the detailed derivation gives the condition \[ r>\sqrt{\frac{89}{72}}\approx 1.112 \] to ensure positive Ricci curvature [2410.01909]. ## 4. Transnormal, isoparametric, and sub-Riemannian structures The Gromoll–Meyer sphere also provides a sharp distinction between transnormal and isoparametric behavior. On \(\mathrm{Sp}(2)\), the function \[ F(Q):=\operatorname{Re}(a),\qquad Q=\begin{pmatrix} a & b\\ c & d \end{pmatrix}\in Sp(2), \] is a properly isoparametric function for a certain left-invariant metric, satisfying \[ |\nabla F|^2 = 1-F^2,\qquad \Delta F=-7F. \] Its focal submanifolds are \[ M_\pm = F^{-1}(\pm1)= \left\{ \begin{pmatrix} \pm1 & 0\\ 0 & d \end{pmatrix} \,\middle|\, d\in S^3 \right\}\cong S^3. \] Because \(F\) is invariant under the \(\star\)-action \[ q \star \begin{pmatrix} a & b\\ c & d \end{pmatrix} = \begin{pmatrix} q & 0\\ 0 & q \end{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} \bar q & 0\\ 0 & 1 \end{pmatrix}, \] it descends to a function \(f\) on \(\Sigma^7\) with \[ |\nabla f|^2=1-f^2. \] However, the projected function is properly transnormal but not isoparametric, with two points as the focal varieties, and the regular level hypersurfaces of \(f\) have non-constant mean curvature [1003.0355]. A different geometric structure arises from principal-bundle horizontality. Using the realization \[ \pi_{GM}:\mathrm{Sp}(2)\to \Sigma^7_{GM}=\mathrm{Sp}(2)/\Delta \] with \(\Delta\cong \mathrm{Sp}(1)\) diagonally embedded in \(\mathrm{Sp}(1)\times \mathrm{Sp}(1)\), the horizontal bundle of the larger \(\mathrm{Sp}(1)\times \mathrm{Sp}(1)\)-bundle over \(S^4\) descends to a rank-\(4\), co-dimension \(3\) distribution \(H^\Sigma\subset T\Sigma^7_{GM}\). The main theorem is that this sub-bundle is completely non-holonomic and of step \(2\) [1608.02444]. In the nested principal-bundle formulation, the same geometry is expressed by \[ {\rm Sp}(2)\xrightarrow{\pi_\Delta}\Sigma_{GM}= {\rm Sp}(2)/\Delta \xrightarrow{\pi} {\rm Sp}(2)/H \cong S^4, \] where \(H={\rm Sp}(1)\times{\rm Sp}(1)\). The induced distribution \[ \mathscr D=d\pi_\Delta(\mathscr D_H) \] on \(\Sigma_{GM}\) is bracket generating of step \(2\), and the submersion \(\pi:\Sigma_{GM}\to S^4\) is an \(S^3\)-bundle over \(S^4\) which is not a principal bundle [2009.00965]. ## 5. Lorentzian and spacetime geometry The commuting-action picture has recently been pushed into semi-Riemannian geometry. Starting from the \(\star\)-diagram with total space \(P=\mathrm{Sp}(2)\), the standard sphere \(\mathrm{S}^7\) and the exotic sphere \(\Sigma^7_{GM}\) are equipped with induced \(\mathrm{S}^1\)-actions. On \(\mathrm{S}^7\subset \mathbb H\oplus \mathbb H\), the action is \[ (a,b)\longmapsto (qa\overline q,qb\overline q), \] for \(q\in \mathrm{S}^1=\{e^{\mathbf i\theta}\}\subset \mathrm{SU}(2)\); on the exotic side, using the parametrization \[ (2\overline cd,|c|^2-|d|^2)\in \mathbb H\oplus \mathbb R, \] the induced action is \[ (2\overline cd,|c|^2-|d|^2)\longmapsto (2q\overline cd\,\overline q,|c|^2-|d|^2). \] A sufficiently negative \(-r^2\)-Cheeger deformation in the orbit direction yields Lorentzian metrics on the regular strata, and the paper calls the resulting structure “almost Lorentzian” when fixed points force degeneration before repair [2403.08960]. The general comparison theorem says that for a \(\star\)-bundle there is a \(\bullet\)-invariant metric on \(M'\) such that \(M/G\) and \(M'/G\) are isometric as metric spaces. Applied to the Gromoll–Meyer diagram, this yields Lorentzian structures on \(\mathrm{S}^7\) and \(\Sigma^7_{GM}\) with the same quotient geometry under the corresponding circle actions. The main corollary states that the classical sphere \(\mathrm{S}^7\) and the Gromoll–Meyer exotic sphere \(\Sigma_{GM}^7\) admit time-oriented Lorentzian metrics of positive Ricci curvature with isometric semi-free actions of \(\mathrm{S}^1\) on an open and dense subset, that these actions fix some points out of the regular stratum, and that the orbit-spaces \(\mathrm{S}^7/\mathrm{S}^1\) and \(\Sigma_{GM}^7/\mathrm{S}^1\) are isometric as metric spaces. The same framework also yields Lorentzian metrics with complete space and time-like geodesics on both manifolds [2403.08960]. ## 6. Extensions, analytic models, and broader significance The Gromoll–Meyer sphere functions as a prototype beyond its own dimension. Pulling back the original \(Sp(2)\to S^7\) principal bundle along suitable equivariant maps produces quotient models for exotic \(8\)- and \(10\)-spheres, and in an orthogonal variant also for Kervaire manifolds and Kervaire spheres. This identifies the \(7\)-dimensional construction as the model case of a more general quotient-of-bundle mechanism rather than a one-off biquotient phenomenon [1010.6039]. It has also supported analytic and probabilistic constructions. In one approach, isometric stochastic flows of a Stratonovich stochastic differential equation are first constructed on the standard sphere \(S^7_s\), realized as \(\mathrm{Sp}(2,\mathbb H)/S^3\) under the \(\bullet\)-action, and then transported to the Gromoll–Meyer sphere \(\Sigma^7_{GM}\), realized as \(\mathrm{Sp}(2,\mathbb H)/S^3\) under the \(\star\)-action, by a homeomorphism \(h:S^7_s\to \Sigma^7_{GM}\). The induced flow, the Fokker–Planck equation, and the entropy functional can then be related closely across the two topological \(7\)-spheres, even though their differential structures are inequivalent [1908.01990]. A related but more indirect connection appears in the theory of special generic maps. Saeki’s theorem states that for \(n\ge 6\) and \(1